This invention relates to methods for producing micro-optical structures, for use as optical elements, such as lenses and gratings, with arbitrary (essentially any) surface-relief profile and, if desired, with optomechanical alignment marks.
Briefly, the invention is carried out by varying the exposure dose, spatially, based upon predetermined contrast curves of a low-contrast photosensitive material. Arbitrary one-dimensional (1-D) or two-dimensional (2-D) surface contours including spherical, aspherical, toroidal, hyperbolic, parabolic, and ellipsoidal can be achieved. The final medium for the fabricated microstructures can be a wide range of materials through the use of etching and replication technology. The method is adapted to mass production. Applications for this invention include, but are not limited to, and particularly the manufacture of microstructures for use in the fields of optical communications, optical data storage, optical interconnects, telecommunications, and in displays, and focal-plane arrays.
This invention is related to the invention described in U.S. patent application, Ser. No. 09/0905,300, filed Jun. 10, 1998 now U.S. Pat. No. 6,075,650, issue Jun. 13, 2000.
A microlens or microcylinder imparts a given phasefront to incident radiation. Similar to macroscopic optical systems, these micro-optical elements can focus incident radiation, diverge the radiation, as well as impart a phase function to correct aberrations from previous elements in the system, or precorrect for aberrations occurring downstream of the microelements. There are numerous methods of fabricating surface-relief micro-optics, and these include multi-step mask and etching, resist thermal transport, e-beam exposure, laser ablation, stamping, etc. The present invention offers advantages over all of these described processes, due to the precision control one has over the final surface shape of the micro-optical element, and in providing optical microstructures having profiles with heights exceeding 15 xcexcm (microns) which in lenses is called a deep sag.
Previous methods of fabrication of elements using exposures of photoresist either concentrated upon fabricating binary (two-level) structures, such as in the case of micro electro-mechanical systems (MEMS), or exposed photoresist to a varying dosage of exposure radiation using thin photoresist. Photoresist has been exposed to create continuous-relief photoresist profiles of optical devices (e.g., microlenses, diffractive phase plates, and diffraction gratings). Exposure methods have included laser pattern generation, see, T. R. Jay and M. B. Stern M. T. Gale, M. Rossi, J. Pedersen, H. Schxc3xctz, xe2x80x9cFabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,xe2x80x9d Opt. Eng. 33, 3556-3566 (1994), grayscale mask exposures, see, H. Andersson, M. Ekberg, S. H{dot over (a)}rd, S. Jacobsson, M. Larsson, and T. Nilsson, xe2x80x9cSingle photomask, multilevel kinoforms in quartz and photoresist: manufacture and evaluation,xe2x80x9d Appl. Opt. 29, pp. 4259-4267 (1990); D. C. O""Shea and W. S. Rockward, xe2x80x9cGray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,xe2x80x9d Appl. Opt. 34, 7507-7517 (1995); D. C. O""Shea and W. S. Rockward, xe2x80x9cGray-scale masks for diffractive-optics fabrication: II. Spatially filtered halftone screens,xe2x80x9d Appl. Opt. 34, 7518-7526 (1995); W. Daschner, P. Long, R. Stein, C. Wu, and S. H. Lee, xe2x80x9cCost-effective mass fabrication of multilevel diffractive optical elements by use of a single optical exposure with a grayscale mask on high-energy beam-sensitive glass,xe2x80x9d Appl. Opt. 36, 4675-4680 (1997), and holographically, M. C. Hutley, xe2x80x9cOptical techniques for the generation of microlens arrays,xe2x80x9d J. of Mod. Opt. 37, 253-265 (1990); H. Ming, Y. Wu, J. Xie, and Toshinori Nakajima, xe2x80x9cFabrication of holographic microlenses using a deep UV lithographed zone plate,xe2x80x9d Appl. Opt. 29, 5111-5114 (1990). But the present invention overcomes limitations of prior processes in affording arbitrary micro-structure profiles, especially with heights (sag in case of lenses) exceeding 15 xcexcm.
In the multi-step mask and etch process, as described in M. B. Stern, xe2x80x9cChapter 3: Binary Optics Fabrication,xe2x80x9d in Micro-Optics: Elements, Systems, and Applications, ed. H. P. Herzig, (Taylor and Francis, Bristol, Pa., 1997), pp. 53-85, one exposes a photosensitive material (typically photoresist developed for the semiconductor industry) using a binary amplitude mask. This mask consists of a series of optically opaque and clear features (e.g., Chromium on glass) that is used to selectively expose selected areas of a photoresist-coated substrate to electromagnetic radiation. After development, one achieves a binary structure in the photoresist that can be transferred into the underlying substrate material through an etching process. This exposure, development and etching process can be repeated with a series of different photomasks in order to achieve a multi-level surface-relief pattern. Although of practical use for the fabrication of diffraction gratings and phase plates, the multi-step masks and etch process has disadvantages when the fabrication of deep-sag or high numerical aperture (NA) micro-optical elements is required. For a diffraction grating fabricated with a multi-level process operating at the order m, the maximum theoretical diffraction efficiency (xcex7m) attainable is given by                                           η            m                    =                                    sin              ⁢                              xe2x80x83                            ⁢                                                c                  2                                ⁡                                  (                                      m                    /                    p                                    )                                                      =                                          [                                                      sin                    ⁡                                          (                                              π                        ⁢                                                  xe2x80x83                                                ⁢                                                  m                          /                          p                                                                    )                                                                            π                    ⁢                                          xe2x80x83                                        ⁢                                          m                      /                      p                                                                      ]                            2                                      ,                            (        1        )            
where p is the number of levels of the structure. Therefore, one notes that if a structure is blazed for 1st order (phase depth of 2xcfx80n and constructed with 16 levels, the diffraction efficiency one can theoretically achieve is 98.7%. Likewise if a structure is blazed for 2nd order (phase depth of 4xcfx80) and constructed with 32 levels, the diffraction efficiency one can achieve is also 98.7%. In other words, if a multi-level profile is such that each level represents xcfx80/8 phase, the efficiency is 98.7%. Similarly, if each level of the structure translates to xcfx80/4 or xcfx80/2 phase, the efficiency drops to 95.0% and 81.1%, respectively. To first-order, one can determine the efficiency of a multi-level microlens structure using Eq. (1) for a diffractive structure. For a surface-relief structure with a refractive index of 1.5 operating at the telecommunications wavelength of 1.3 xcexcm, a physical depth of 2.6 xcexcm corresponds to a phase depth of 2xcfx80 according to                               φ          =                                                    2                ⁢                π                            λ                        ⁢                          d              ⁡                              (                                  n                  -                  1                                )                                                    ,                            (        2        )            
where xcfx86 is the phase depth, d is the relief depth, xcex is the operating wavelength and n is the index of refraction of the substrate material (air is assumed to be the second medium). Therefore, if one has a microlens that requires a sag of 10.4 xcexcm, the phase depth is 8 xcexcn. With a 16-level structure, the element will only be 81% efficient at best (assuming no fabrication errors).
Such a microlens with additional levels in order to increase the optical efficiency, is impractical to attain. For a 99 percent efficient f/3 microlens operating at xcex=1.3 xcexcm, the critical dimension (CD) of the surface features required is approximately 0.5 xcexcm. This CD offers an extreme challenge in terms of achieving such features, let alone the mask-to-mask alignment tolerances required (generally one quarter of the CD). Since microlenses required for telecommunications and optical data storage can require f/2 and f/1 optical speeds, the use of multi-level mask and etch technology is therefore impractical for achieving the majority of the optical elements required.
Laser pattern generators, which have been proposed, have emphasized the production of binary masks. See, for example, S. Charles Baber, xe2x80x9cApplication of high resolution laser writers to computer generated holograms and binary diffractive optics,xe2x80x9d Holographic Optics: Optically and Computer Generated, SPIE Proc.1052 66-76 (1989); E. Jxc3xa4ger, J. Hoxcex2feld, Q. Tang, T. Tschudi, xe2x80x9cDesign of a laser scanner for kinofrom fabrication,xe2x80x9d Holographic Optics II: Principles and Applications, SPIE Proc. 1136, 228-235 (1989). Laser pattern generators for making diffractive kinoforms are described in M. T. Gale, M. Rossi, J. Pedersen, H. Schxc3xctz, xe2x80x9cFabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,xe2x80x9d Opt. Eng. 33, 3556-3566 (1994).
Laser pattern generators have been proposed to expose photoresist in a point-by-point fashion with variable exposure doses, see, for example, M. T. Gale, M. Rossi, J. Pedersen, H. Schxc3xctz, xe2x80x9cFabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,xe2x80x9d Opt. Eng. 33, 3556-3566 (1994) to achieve continuous-relief microstructures. Likewise, proposed has been the use of grayscale mask lithography to produce continuous-relief profiles in photoresist, see Vlannes, U.S. Pat. No. 5,004,673; G. Gal, U.S. Pat. No. 5,310,623, and a U.S. Pat. No. 5,482,000 to G. Gal.
The present invention improves upon prior LPG methods even for the fabrication of diffractive elements, by patterning of photoresist with a blazed multi-level or continuous profiles and with depths exceeding 15 xcexcm. Prior to the present invention, no one had fabricated continuous-relief microlenses using a variable exposure dosage that resulted in microstructures with surface sags exceeding 15 xcexcm.
Using thermal transport mechanisms to fabricate micro-optical components, one takes a pre-patterned surface relief structure, and through the introduction of heat above the thermal glass transition temperature of the material (Tg) the structure is melted into a spherical or cylindrical profile by surface-tension effects. Initially most thermal transport work was performed with binary patterned photoresist, as described in Z. D. Popovic, R. A. Sprague, and G. A. Neville Connell, xe2x80x9cTechnique for monolithic fabrication of microlens arrays,xe2x80x9d Appl. Opt. 27, 1281-1284 (1988); D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, xe2x80x9cThe manufacture of microlenses by melting photoresist,xe2x80x9d Meas. Sci. Technol. 1, 759-766 (1990). One first exposes a substrate with photoresist, and through a binary high-contrast exposure process, patterns the photoresist into a series of cylindrical surface-relief structures. By heating the substrate using a convection oven or hot plate the photoresist can be melted. Due to surface tension, the resulting melted profile takes the form of a spherical surface. This process is conducive to the fabrication of microlenses in the f/1 to f/3 range, and is not conducive towards slower microlenses due to the tendency of the melted photoresist to sag in the middle.
The continuous-relief profile of the photoresist can be transferred into the underlying substrate material through a dry etching process, see, E. J. Gratrix, xe2x80x9cEvolution of a microlens surface under etching conditions,xe2x80x9d Proc. SPIE 1992, pp. 266-274 (1993); M. B. Stern and T. R. Jay, xe2x80x9cDry etching for coherent refractive microlens arrays,xe2x80x9d Appl. Opt. 33, pp. 3547-3551 (1993). In recent years, more sophisticated methods of producing microlenses through the thermal transport scheme have been proposed. Hlinka et al, U.S. Pat. No. 5,718,830 proposed using photoresist masking and RIE etching to produce cylindrical islands in a PMMA layer that is spun onto a substrate. These cylindrical islands are then melted in order to produced the desired microlens in PMMA. A similar concept was filed in 1994 (Iwasaki et al, U.S. Pat. No. 5,298,366) that covered the use of an intermediate layer in order to transfer the cylindrical islands into a final material other than photoresist. This second material could then be thermally reflowed in order to produce the desired microlenses. Feldblum et al, U.S. Pat. No. 5,286,338 realized the deficiency of the thermal transport mechanism to produce microlenses with precision surfaces for diffraction-limited optical performance, and proposed the use of reactive ion etching to rectify the situation. One can take resist microlenses with precision surfaces for diffraction-limited optical performance, and proposed the use of reactive ion etching to rectify the situation. One can take resist microlenses produced using the thermal transport mechanism, and etch-transfer them into the underlying substrate using reactive ion etching (RIE). By changing the gas constituency of the RIE chamber during etch-transfer, one can change the etch selectivity (ratio of substrate etch rate to photoresist etch rate), and thereby modify the profile of the resultant microlens to be closer to the desired profile than the thermal transport method alone can achieve.
Another method which has been proposed for improving the profile obtained by the thermal transfer method involves preshaping the microlens structure before melting have been implemented, including multistep, as described in D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, xe2x80x9cThe manufacture of microlenses by melting photoresist,xe2x80x9d Meas. Sci. Technol. 1, 759-766 (1990), and laser pattern generation, as described in T. Jay and M. Stern, xe2x80x9cPreshaping photoresist for refractive microlens fabrication,xe2x80x9d Opt. Eng. 33, 3552-3555 (1994). Once preshaped, the photoresist structure is then melted to smooth the surface-relief profile.
Researchers have also patterned non-photoresist materials such as InP. GaAs, and GaP, using the thermal transport method as described in Z. L. Liau, V. Diodiuk, J. N. Walpole, and D. E. Mull, xe2x80x9cLarge-numerical-aperture InP lenslets by mass transport,xe2x80x9d Appl. Phys. Lett. 52, 1859-1861 (1988); Z. L. Liau, D. E. Mull, C. L. Dennis, R. C. Williaamson, and R. G. Waarts, xe2x80x9cLarge-numerical-aperture microlens fabrication by one-step etching and mass-transport smoothing,xe2x80x9d Appl. Phys. Lett. 64, 1484-1486 (1994); J. S. Swenson, Jr., R. A. Field, L. Mitt. Abraham, xe2x80x9cEnhanced mass-transport smoothing of f/0.7 GaP microlenses by use of sealed ampoulesxe2x80x9d, Appl Phys. Lett. 66 1304-1306 (1995). For the non-polymer materials, one requires several days of a controlled heating processes in special atmospheres wherein oven temperatures can reach 1000xc2x0 C. For the photoresist-melting method of fabricating microstructures, the temperatures are generally in the 160-200xc2x0 C. The transfer of the continuous-relief profile of the photoresist can be transferred into the underlying substrate material through a dry etching process, see, E. J. Gratrix, xe2x80x9cEvolution of a microlens surface under etching conditions,xe2x80x9d Proc. SPIE 1992, pp. 266-274 (1993); M. B. Stern and T. R. Jay, xe2x80x9cDry etching for coherent refractive microlens arrays,xe2x80x9d Appl. Opt. 33, pp. 3547-3551 (1993).
A disadvantage of the thermal transport mechanism is the ability to achieve microlens arrays with a high filling factor (ratio of area taken up by the optical elements to the total area of the substrate). One requires a space between the patterned pillar area of the resist, PMMA, or other material in order for distinct structures to be defined that can be melted into shape. If the filling factor is less than 100%, this can reduce the overall efficiency of the optical system. One solution proposed by Aoyama and Shinohara, in U.S. Pat. No. 5,694,246 is to first pattern a sparse array so that the separation of microlenses is twice a given microlens diameter. After melting and fixing the material, the sparse microlens array is recoated with photoresist and patterned with a second sparse array, but this one aligned such that the cylindrical islands of the second array are positioned between the gaps of the microlenses of the first array. The second array, is then reflowed in order to produce a single microlens array with close to 100% filling factor. The disadvantages of the method proposed by Aoyama and Shinohara are the extra processing steps, and the additional precision one requires in aligning one array to a second one.
A principal disadvantage of the thermal transport method is the lack of control that one has regarding the surface-relief profile. Since the final processing step involves a thermal reflow, the shape and quality of any alignment marks simultaneously patterned onto the surface is severely limited. Therefore, sharp registration features with submicron accuracies have not been achieved.
The invention also allows for microstructures to have filling factors up to 100% without any additional processing steps. The invention enables one to simultaneously pattern alignment or registration marks. With the thermal reflow method, the shape and quality of any alignment marks simultaneously patterned onto the surface is severely limited. Therefore, sharp registration features with submicron accuracies are not possible using this thermal transport technique.
Other methods of producing microlenses have concentrated upon fabricating a microlens directly onto an optical fiber (see, for example, Edwards et al, U.S. Pat. No. 5,011,254, and Modavis and Webb, U.S. Pat. No. 5,455,879). These techniques typically involve drawing the end of the fiber, or otherwise tapering it in such a manner as to produce a surface capable of at least partially collimating or focusing the radiation being emitted by the fiber. The invention described herein does not relate to such structures. Rather it relates to a microstructured substrate that is independently fabricated (but then may be integrated onto the end of a fiber if required). The preforming of glass has been proposed to create stand-alone glass microcylinders. Snyder and Baer, U.S. Pat. No. 5,155,631, describes preforming glass such that a cylindrical lens is produced of arbitrary curvature. This cylindrical lens can be integrated into an optical system in order to collimate laser diode arrays. The present invention differs from the process described by Snyder and Baer, in that the present invention does not involve the use of heat to preform materials, and is not limited to cylindrical microstructures.
It has been proposed to use grayscale contact prints or projection printing to fabricate microstructures in photoresist. See, H. Andersson, M. Ekberg, S. H{dot over (a)}rd, S. Jacobsson, M. Larsson, and T. Nilsson, xe2x80x9cSingle photomask, multilevel kinoforms in quartz and photoresist: manufacture and evaluation,xe2x80x9d Appl. Opt. 29, pp. 4259-4267 (1990); D. C. O""Shea and W. S. Rockward, xe2x80x9cGray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,xe2x80x9d Appl. Opt. 34, 7507-7517 (1995); D. C. O""Shea and W. S. Rockward, xe2x80x9cGray-scale masks for diffractive-optics fabrication: II. Spatially filtered halftone screens,xe2x80x9d Appl. Opt. 34, 7518-7526 (1995); W. Daschner, P. Long, R. Stein, C. Wu, and S. H. Lee, xe2x80x9cCost-effective mass fabrication of multilevel diffractive optical elements by use of a single optical exposure with a grayscale mask on high-energy beam-sensitive glass,xe2x80x9d Appl. Opt. 36, 4675-4680 (1997). See also, M. C. Hutley, xe2x80x9cChapter 5: Refractive Lenslet Arrays,xe2x80x9d in Micro-Optics: Elements, Systems, and Applications, ed. by H. P. Herzig, (Taylor and Francis, Bristol,. Pa., 1997), pp. 127-150. The grayscale masks can also be produced using photographic slide film or through the use of high-energy beam sensitive (HEBS) glass plates. But such proposals have not been practical when micro-structure with arbitrary profiles and especially height or sags exceeding 15 xcexcm are needed.
The fabrication method provided by the current invention enables the direct patterning of photosensitive material in accordance with a variable dose of electromagnetic radiation. The photoresist coating, exposure, and development process enable one to create surface-relief profiles of arbitrary surface micro-structure. The previous methods described above have not provided optical micro-structures having the arbitrary relief profiles and, if desired, with alignment and registration marks capable of the fabrication process of the present invention. The invention enables to the use of low-contrast photosensitive material to achieve a final structure (replicated, etch-transferred, etc.) that has a profile height (surface sag) greater than 15 xcexcm.
In accordance with the present invention, photosensitive material is exposed to a spatially variable dose of electromagnetic energy to create a surface-relief structure upon development of the photosensitive material. The photosensitive material may be coated, onto a substrate of interest (planar or otherwise). The coating is characterized by its response curve in terms of developed relief depth to electromagnetic exposure energy and wavelength.
The invention involves the recognition that the response curve is a complex function of the material parameters as well as the method with which the coating, exposure, and development process steps are performed. For instance, the material""s viscosity, in conjunction with the coating parameters (spin speed in the case of spin-coating, pull rate in the case of dip coating, etc.), will dictate the final film thickness. During exposure, the wavelength of the radiation used, in conjunction with pre-exposure procedures (such as the temperature and duration of an oven bake), complex index of refraction of the photosensitive material, and chemical compound of the material being exposed, are parameters that will dictate the sensitivity of the material, and therefore rate of development. Development procedures can also vary the response curve. Development time and development solution used also affect development rates, but so will the exact method of development. In the case of aqueous development, the response curve will change if one uses immersion, spray, or puddle methods of developing the photosensitive material. The invention is carried out by selecting, the photosensitive materials and controlling the coating, exposure, and development parameters, to achieve the precision microstructures desired.
The resulting microstructure in the photosensitive material can remain in the material, or be etch-transferred into the underlying substrate. The microstructure can also be replicated into a polymer material via a cast-and-cure, embossing, compression molding, or compression injection molding process. This enables mass production of the optical micro-structures. The disclosed manufacturing process is robust in that arbitrary surface-relief structures can be fabricated that have optical and mechanical properties of interest. Optical micro-structures of particular interest include microlenses with toroidal, elliptical, and hyperbolic surface-relief structures.
Preferably, laser pattern generation (LPG) is practiced in-accordance with this invention. One exposes photoresist using a single or multiple focused laser beam that rasters across a photosensitive substrate. There are two scanning geometries that are generally preferred: x-y scanning, where the substrate is moved on a pair of orthogonal linear stages, and r-xcex8 scanning, where the substrate is spun on an air bearing spindle. In the x-y scan systems, the part is scanned below a single or multiple focused laser beams. For photoresist that is sensitive in the blue or UV portions of the electromagnetic spectrum, the radiation source used is typically an argon-ion, krypton-ion, or helium-cadium laser. Semiconductor laser diodes may be used for LPGs. The invention is not limited to the source of exposure energy and other sources of electromagnetic radiation such as the LED or electron beams may be used, lasers however are now preferred. To spatially vary the exposure dose the photosensitive material is exposed to, one can vary the speed of the stages moving the focused exposure beams or the substrate. This changes the dwell time of the exposure beam. A more accurate method of controlling the exposure dosage is to vary the intensity of the writing beam or beams. Methods for accomplishing this include the use of an electro-optic or acousto-optic modulators with diode lasers, including those which may provide radiation in the blue or UV portion of the electromagnetic spectrum, the drive current of the laser can be directly modulated to vary the outputed laser beam""s power. A computer desirably is used to control the modulator and computes, based upon the desired surface-relief pattern and the response curve of the photosensitive material, the modulation sequence required. After the relief pattern has been developed, the element can be used as is, or one can transfer the desired pattern into the substrate using an etch process. The patterned surface can also be used as the master element for a replication process.
With the gray-scale mask lithography method, practiced with this invention, one can expose photoresist using a mask with a transmission function T(x,y). By passing a beam of uniform or well-defined intensity variation Iinc(x,y) through this mask, the transmitted beam can have a controlled intensity function Iout(x,y)=Iinc(x,y) T(x,y). The intensity function can be used to expose photoresist, or any other photosensitive material once one has a well-characterized response curve for the material.
Other exposure methods of capitalizing upon this invention include the use of moving apertures and controlled diffraction effects in order to achieve the surface-relief profiles desired. With the moving aperture method, an amplitude mask is translated in front of an electromagnetic exposure beam. By choosing the amplitude distribution of the mask and the path with which the mask and/or substrate is being translated, one can control the spatial distribution of the exposure dose for a particular photosensitive material.
Another method of achieving desired surface-relief profiles is to expose a photosensitive material to the intensity of a diffracted electromagnetic (such as an optical) beam. The diffraction profile can be achieved by controlling the temporal and spatial coherence of the exposure beam, controlling the aperture shape the radiation is diffracting around, as well as the distance from the aperture the photosensitive material is set at. For instance, one can diffract or focus a beam through a pinhole in order to create an exposure dosage that is circularly symmetric but decreases radially. By tailoring the exposure profile to the response curve of the photosensitive material (and vice-versa) one can then write and develop a microlens structure. One can then translate the substrate underneath the radially symmetric exposure dose in order to achieve a cylindrical microlens.
The current invention allows one to expose arbitrary profiles. Once the response curve of a photosensitive material is characterized, then the exposure methods, such as mentioned above may be used to create the relief structures dictated by a particular optical or mechanical design. For optical applications, phase formats of interest include radially symmetric profiles, as well as non-rotationally symmetric profiles. Ray tracing programs such as Optical Research Associate""s (ORA""s) Code V and Sinclair Optics OSLO SIX, have different conventions in terms of how the phase of a diffractive optical element (DOE) is represented, so only the conventions used by Code V will be given for illustrative purposes. For rotationally symmetric phase plates, the phase polynomial xcfx86 (r)can be represented by
xcfx86(r)=c1r2+c2r4+c3r6+c4r8+xe2x80x83xe2x80x83(3)
To better relate such a rotationally symmetric phase profile to that of a conic constant, Code V also allows for input of the phase function according to                                           φ            ⁡                          (              r              )                                =                                                    cr                2                                            1                +                                                      1                    -                                                                  (                                                  1                          +                          k                                                )                                            ⁢                                              c                        2                                            ⁢                                              r                        2                                                                                                                  +                          Ar              4                        +                          Br              6                        +                          Cr              8                        +            …                          ⁢                  xe2x80x83                ,                            (        4        )            
where k is the conic constant, and c is the curvature at the pole of the surface. For a pure conic surface, the coefficients A, B, C, etc. are zero. For a sphere, k=0, for a hyperboloid, k less than xe2x88x921, for a paraboloid, k=xe2x88x921, and for an ellipsoid, xe2x88x921 less than k less than 0.
For non-rotationally symmetric profiles, one can describe the phase function required using
xcfx86(x,y)=c1x+c2y+c3x2+c4xy+c+c5y2+xe2x80x83xe2x80x83(5)
Equations (3) through (5) represent the standard phase inputs of Code V that any arbitrary phase function can be fit to. For instance, one may desire a phase function that has Zernike polynomials added to the phase structure in order to improve the micro-optical element""s tolerance to alignment errors. Other phase terms can be added to the relief profile that act to compensate or pre-compensate for other elements in the optical system. In addition to describing the phase of the optical element, one can separately define in a more graphical way alignment or registration marks that are probed interferometrically, mechanically, or through optical imaging in order to align the micro-element.
The manufacturing process provides benefits for telecommunications and optical data storage by enabling the fabrication of optical elements for optical interconnects and for communications.